# Questions tagged [cauchy-schwarz-inequality]

The Cauchy-Schwarz inequality states $|\langle x,y \rangle |\leq ||x||\cdot ||y||.$ Use this tag for questions related to the CS inequality and its applications.

23
questions

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votes

**1**answer

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### Tight sublinear estimates for a triple partial binomial summation

Is there tight estimates for the following logarithmic summation ($\gamma,\gamma'\in(0,1)$ and $\mu,\mu'>0$)
$$\log_2\Bigg(\sum_{t=\frac{n^{}}2-n^\gamma\sqrt{\mu\ln n}}^{\frac{n^{}}2+n^\gamma\sqrt{...

**1**

vote

**1**answer

97 views

### Tight estimates for binomial summation

Is there tight estimates for the following logarithmic summation ($\gamma\in(0,1)$)
$$\ln\Bigg(\sum_{t=\frac{n^{}}2-\gamma n^\gamma}^{\frac{n^{}}2+\gamma n^\gamma}\sum_{\ell=\frac{n^{}}2-\gamma n^\...

**3**

votes

**2**answers

366 views

### Good upper bound for a certain sum

Given $\gamma \in [0, 1)$, an integer $N \ge 2$, and a decreasing null sequence of positive numbers $e_1,e_2,\ldots,e_t,\ldots$, I'm interested in estimating the sum $S_N := \sum_{t=1}^N\gamma^t e_{N-...

**5**

votes

**1**answer

150 views

### Good upper-bound for $\mathbb E[|X-np|^r]$ where $X \sim \text{Binomial}(n,p)$ and $r \ge 1$

Disclaimer. Question moved from SE.
Setup
Let $X \sim \text{Binomial}(p, n)$, and $r \ge 1$.
Question
What is a good upper-bound for $\mathbb E[|X-np|^r]$ ?
Solution for small $r$
If $r=2$, then ...

**2**

votes

**0**answers

50 views

### Bounds for $\sum_{t=1}^Tn_t(s_t)^{-\alpha}\mu(s_t)$ where $n_t(s) = \sum_{1 \le t' \le t} 1_{\{s_{t'}=s\}}$ for $s \in [k]$ and $\mu \in \Delta_k$

Disclaimer: I'm not certain this is the right venue for this post, but I'll give it a try...
So trying prove some bounds in my ongoing work in theoretical reinforcement learning, I encountered the ...

**1**

vote

**1**answer

265 views

### Is there a tight lower bound for the expectation of the product of two positive valued random variables?

Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$.
I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely.
...

**0**

votes

**1**answer

86 views

### Inequality involving product-of-minus vs minus-of-product for positive integers

I'm encountering this inequality for dimensionality reduction problem. The simplified form looks as follows:
Consider positive integers $a_1$, $a_2$, $b_1$ and $b_2$ where $a_1>b_1$ and $a_2>...

**3**

votes

**1**answer

288 views

### A moment inequality

Let $\chi(s)=\int_{0}^{1}x(t)^{s}f(t)dt$,
where $x(t)$ and $f(t)$ are real valued continuous functions for
$t\in[0,1]$, and $f(t)\geq0$.
Is it possible to show that
$\left(\chi(0)\chi(2)-\chi(1)^{2}...

**2**

votes

**1**answer

164 views

### Is it possible to find $f$ such that : $f$ is absolutely integrable, $f'$ is absolutely integrable and such that $f$ is not $1/2$-Hölder

I am trying to find a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ that fullfils the following conditions
$$f \in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R}^+)$$
$$\int_{\mathbb{R}^+} f \in \...

**-2**

votes

**1**answer

223 views

### On the Cauchy-Schwarz Inequality for trace function of random matrices

In the deterministic case, for two matrices $A$ and $B$ with appropriate matrices, we know that
$$tr((A^{T}B)^{2})\leq tr(A^{T}A)tr(B^{T}B)$$
which is the trace form of Cauchy-Schwarz-Inequality (CSI)....

**12**

votes

**3**answers

473 views

### A probabilistic angle inequality

Conjecture: There is a universal constant $c$ such that for any fixed nonzero real vector $q$ of any dimension $n$ and any random vector $p$ of the same dimension $n$ with independent components ...

**5**

votes

**1**answer

334 views

### An inequality involving a sum of power terms

I am currently working in a problem in Information Theory and I came across a difficult inequality. After many attemps, I simplified the inequality, which now looks at follows.
Consider a positive ...

**2**

votes

**2**answers

359 views

### An alternative proof of Bayesian Cramer-Rao

My question is:
Are there an alternative proof of Cramer-Rao lower bound that does not use
Cauchy-Swartz inequality?
Let me outline the classical proof and explain why I am interested in this ...

**0**

votes

**1**answer

122 views

### Upper bound of $\frac{\sum_i c_ia_ie_i}{\sum_i d_ib_if_i}$?

Let $\sum_i c_i =\sum_i d_i=1$, where $c_i,d_i \ge 0$. Assume that $\frac{\sum_i c_ia_i}{\sum_i d_ib_i} \le \epsilon_1$ and $\frac{\sum_i c_ie_i}{\sum_i d_if_i} \le \epsilon_2$, where $a_i,b_i,e_i,f_i ...

**4**

votes

**0**answers

221 views

### Does the Cauchy–Schwarz inequality imply 2-positivity?

Recall the following generalisation of Cauchy–Schwarz.
Theorem. Let $f\colon \mathscr{A} \to \mathscr{B}$ be a linear 2-positive map between C$^*$-algebras. Then for all $a,b \in \mathscr{A}$ we ...

**2**

votes

**1**answer

127 views

### Majorization of cyclic products

Let $k,m,n\in\mathbb N$ such that $n>k$. For a partition $\alpha=(\alpha_1,\dots,\alpha_k)\vdash m$ with $\alpha_1\ge\dots\ge \alpha_k>0$ and nonnegative $ x_1,\dots,x_n$ define $x^\alpha :=\...

**2**

votes

**2**answers

381 views

### Does this simple inequality have a name?

Let $x_{1},\ldots,x_{n}$ be nonnegative numbers such that $m \leq x_{i} \leq M$. Let
$$
S=\sum_{i=1}^{n}{x_{i}}
$$
and
$$
Q=\sum_{i=1}^{n}{x_{i}^{2}}.
$$
Then
$$
Q \leq S(M+m)-nMm.
$$
This has ...

**5**

votes

**0**answers

1k views

### A stronger Cauchy-Schwarz inequality for traces of compression matrices

Assume that $A$ and $B$ are contractions, so
$I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let
$C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that:
$$Tr\left(\frac{1}{1-AA^T}\right)...

**4**

votes

**1**answer

211 views

### A homogeneous but slightly asymmetric inequality

I need to prove the following inequality: for any $Z=(z_1,\dots,z_l)\in\mathbb{C}^l$ for any $p\geq 2$ and $l\geq 2$
\begin{equation}
\left|\left|\sum_{j=1}^l z_j\right|^p-\sum_{j=1}^l\left|z_j\right|^...

**6**

votes

**3**answers

2k views

### Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space

I previously posted this question on Math.SE but didn't receive an answer. It is perhaps a little vague; part of what I want to know is what question I should ask.
First, consider the following form ...

**29**

votes

**1**answer

2k views

### Can the fact that the square of an integer is a natural number be categorified?

If $a$ and $b$ are natural numbers, then $a-b$ is an integer and so the square $(a-b)^2$ is a natural number. In particular
$$ (a-b)^2 \geq 0. \qquad (1)$$
Combining this fact with the identity
$...

**2**

votes

**3**answers

573 views

### An Linear Algebra Inequality

How to prove the following inequality:
Let $X$ and $Y$ be $n\times m$ matrices with real entries. Prove that
\begin{equation}
\det\left(XY^T\right)^2 \leq \det\left(XX^T\right)\det\left(YY^T\right) .
\...

**8**

votes

**5**answers

2k views

### A plausible inequality

I come across the following problem in my study.
Let $x_i, y_i\in \mathbb{R}, i=1,2,\cdots,n$ with $\sum\limits_{i=1}^nx_i^2=\sum\limits_{i=1}^ny_i^2=1$, and $a_1\ge a_2\ge \cdots \ge a_n>0 $. Is ...